Conditional Analysis on Rd

Type of Publication: Contribution to a collection
Publication status: Published
Author: Cheridito, Patrick; Kupper, Michael; Vogelpoth, Nicolas
Year of publication: 2015
Published in: Set optimization and applications - the state of the art : from set relations to set-valued risk measures / Hamel, Andreas H. et al. (ed.). - Berlin [u.a.] : Springer, 2015. - (Springer Proceedings in Mathematics & Statistics ; 151). - pp. 179-211. - ISBN 978-3-662-48668-9
DOI (citable link): https://dx.doi.org/10.1007/978-3-662-48670-2_6
Summary:
This paper provides versions of classical results from linear algebra, real analysis and convex analysis in a free module of finite rank over the ring L0 of measurable functions on a σ-finite measure space. We study the question whether a submodule is finitely generated and introduce the more general concepts of L0-affine sets, L0-convex sets, L0-convex cones, L0-hyperplanes and L0-halfspaces. We investigate orthogonal complements, orthogonal decompositions and the existence of orthonormal bases. We also study L0-linear, L0-affine, L0-convex and L0-sublinear functions and introduce notions of continuity, differentiability, directional derivatives and subgradients. We use a conditional version of the Bolzano–Weierstrass theorem to show that conditional Cauchy sequences converge and give conditions under which conditional optimization problems have optimal solutions. We prove results on the separation of L0-convex sets by L0-hyperplanes and study L0-convex conjugate functions. We provide a result on the existence of L0-subgradients of L0-convex functions, prove a conditional version of the Fenchel–Moreau theorem and study conditional inf-convolutions.
Subject (DDC): 510 Mathematics
Bibliography of Konstanz: Yes

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CHERIDITO, Patrick, Michael KUPPER, Nicolas VOGELPOTH, 2015. Conditional Analysis on Rd. In: HAMEL, Andreas H., ed. and others. Set optimization and applications - the state of the art : from set relations to set-valued risk measures. Berlin [u.a.]:Springer, pp. 179-211. ISBN 978-3-662-48668-9. Available under: doi: 10.1007/978-3-662-48670-2_6

@incollection{Cheridito2015-11-18Condi-33461, title={Conditional Analysis on Rd}, year={2015}, doi={10.1007/978-3-662-48670-2_6}, number={151}, isbn={978-3-662-48668-9}, address={Berlin [u.a.]}, publisher={Springer}, series={Springer Proceedings in Mathematics & Statistics}, booktitle={Set optimization and applications - the state of the art : from set relations to set-valued risk measures}, pages={179--211}, editor={Hamel, Andreas H.}, author={Cheridito, Patrick and Kupper, Michael and Vogelpoth, Nicolas} }

2016-03-24T10:03:03Z Kupper, Michael Cheridito, Patrick Vogelpoth, Nicolas Vogelpoth, Nicolas eng 2015-11-18 2016-03-24T10:03:03Z Conditional Analysis on R<sup>d</sup> Cheridito, Patrick This paper provides versions of classical results from linear algebra, real analysis and convex analysis in a free module of finite rank over the ring L<sup>0</sup> of measurable functions on a σ-finite measure space. We study the question whether a submodule is finitely generated and introduce the more general concepts of L<sup>0</sup>-affine sets, L<sup>0</sup>-convex sets, L<sup>0</sup>-convex cones, L<sup>0</sup>-hyperplanes and L<sup>0</sup>-halfspaces. We investigate orthogonal complements, orthogonal decompositions and the existence of orthonormal bases. We also study L<sup>0</sup>-linear, L<sup>0</sup>-affine, L<sup>0</sup>-convex and L<sup>0</sup>-sublinear functions and introduce notions of continuity, differentiability, directional derivatives and subgradients. We use a conditional version of the Bolzano–Weierstrass theorem to show that conditional Cauchy sequences converge and give conditions under which conditional optimization problems have optimal solutions. We prove results on the separation of L<sup>0</sup>-convex sets by L<sup>0</sup>-hyperplanes and study L<sup>0</sup>-convex conjugate functions. We provide a result on the existence of L<sup>0</sup>-subgradients of L<sup>0</sup>-convex functions, prove a conditional version of the Fenchel–Moreau theorem and study conditional inf-convolutions. Kupper, Michael

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